A Vector Equilibrium Problem for Symmetrically Located Point Charges on a Sphere

We study the equilibrium measure on the two-dimensional sphere in the presence of an external field generated by r + 1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we chara...

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Bibliographic Details
Published in:Constructive approximation 2022-06, Vol.55 (3), p.775-827
Main Authors: Criado del Rey, Juan G., Kuijlaars, Arno B. J.
Format: Article
Language:English
Subjects:
Analysis
Droplets
Mathematics
Mathematics and Statistics
North Pole
Numerical Analysis
Riemann surfaces
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Summary:We study the equilibrium measure on the two-dimensional sphere in the presence of an external field generated by r + 1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or the full half-line. The two interval case corresponds to a droplet with two disjoint components, and it is analyzed by means of a genus one Riemann surface.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-022-09566-5